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 A001661 Largest number not the sum of distinct positive n-th powers. (Formerly M5393 N2342) 17
 128, 12758, 5134240, 67898771, 11146309947, 766834015734, 4968618780985762 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS a(8) > 74^8. - Donovan Johnson, Nov 23 2010 Fuller and Nichols prove that a(6) = 11146309947 and that 2037573096 positive numbers cannot be written as the sum of distinct 6th powers. - Robert Nichols, Sep 09 2017 a(8) >= 83^8 ~ 2.25e15 since A030052(8) = 84. Similarly, a(9..15) >= (46^9, 62^10, 67^11, 80^12, 101^13, 94^14, 103^15) ~ (9.2e14, 8.4e17, 1.2e20, 6.9e22, 1.1e26, 4.2e27, 1.6e30), cf. formula. Most often a(n) will be closer to and even larger than A030052(n)^n. - In the literature, a(n)+1 is known as the anti-Waring number N(n,1). - M. F. Hasler, May 15 2020 a(9..16) > (1.55e17, 1.31e19, 1.64e21, 5.55e23, 1.32e26, 1.37e28, 2.09e30, 9.99e35). - Michael J. Wiener, Jun 10 2023 REFERENCES S. Lin, Computer experiments on sequences which form integral bases, pp. 365-370 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970. Harry L. Nelson, The Partition Problem, J. Rec. Math., 20 (1988), 315-316. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Table of n, a(n) for n=2..8. R. E. Dressler and T. Parker, 12,758, Math. Comp. 28 (1974), 313-314. Shalosh B. Ekhad and Doron Zeilberger, Automating John P. D'Angelo's method to study Complete Polynomial Sequences, arXiv:2111.02832 [math.NT], 2021. Mauro Fiorentini, Rappresentazione di interi come somma di potenze (in Italian). C. Fuller and R. H. Nichols Jr., Generalized Anti-Waring Numbers, J. Int. Seq. 18 (2015), #15.10.5. R. L. Graham, Complete sequences of polynomial values, Duke Math. J. 31 (1964), pp. 275-285. D. Kim, On the largest integer that is not a sum of distinct nth powers of positive integers, arXiv:1610.02439 [math.NT], 2016-2017. D. Kim, On the largest integer that is not a sum of distinct nth powers of positive integers, J. Int. Seq. 20 (2017), #17.7.5. P. LeVan and D. Prier, Improved Bounds on the Anti-Waring Number, J. Int. Seq. 20 (2017), #17.8.7. D. C. Mayer, Sharp bounds for the partition function of integer sequences, BIT 27 (1987), 98-110. D. C. Mayer, Partition functions via bit list operations, 2009. N. J. A. Sloane and R. E. Dressler, Correspondence, June 1974 R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z. 51 (1948) 466-468. Eric Weisstein's World of Mathematics, Waring's Problem M. J. Wiener, The Largest Integer Not the Sum of Distinct 8th Powers, J. Integer Sequences, 26 (2023), Article 23.5.4. J. W. Wrench, Jr., Letter to N. J. A. Sloane, 10 Apr, 1974 FORMULA a(n) < d*2^(n-1)*(c*2^n + (2/3)*d*(4^n - 1) + 2*d - 2)^n + c*d, where c = n!*2^(n^2) and d = 2^(n^2 + 2*n)*c^(n-1) - 1, according to Kim [2016-2017]. - Danny Rorabaugh, Oct 11 2016 a(n) >= (A030052(n)-1)^n. - M. F. Hasler, May 15 2020 CROSSREFS Cf. A030052, A173563, A279529. Cf. A121571 (primes instead of integers). Sequence in context: A282398 A297101 A181250 * A264095 A239190 A283856 Adjacent sequences: A001658 A001659 A001660 * A001662 A001663 A001664 KEYWORD nonn,nice,more,hard AUTHOR N. J. A. Sloane and Robert G. Wilson v EXTENSIONS a(7) from Donovan Johnson, Nov 23 2010 a(8) from Michael J. Wiener, Jun 10 2023 STATUS approved

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Last modified September 24 16:22 EDT 2023. Contains 365579 sequences. (Running on oeis4.)